(1) Field of the Invention
This invention pertains to the field of magnetic resonance imaging and multi-dimensional image reconstruction from partially acquired data. More particularly, this invention pertains to a method of reconstructing a complex MRI image from a portion of asymmetrically acquired data using an iterative multidimensional inverse transform technique. The use of the method reduces data acquisition time by reducing the amount of data needed to reconstruct the image matrix.
(2) Description of the Related Art
Acquiring data in many imaging modalities often takes a significant length of time. Typically, as less
Acquiring data in many imaging modalities often takes a significant length of time. Typically, as less of the data are acquired, the resolution of the image is decreased while the temporal resolution is increased. On the other hand, the collection of more data allows for higher resolution but requires longer acquisition times.
The size of the data matrix used to generate an image is known to affect the resolution. In one dimension, if the data matrix coverage comprises data from xe2x88x92kmax to kmax, then the resolution is given by xcex94x=1/(2kmax). Many methods of reducing image acquisition time are based on the theory that it is not necessary to collect all the data for the data matrix since, theoretically, the data is purely real and will therefore express complex conjugation symmetry. By using complex conjugation symmetry, the negative terms in a data matrix s(k) can be generated from the positive terms since s(xe2x88x92k)=s*(k). Therefore, if only half of the data of a given data matrix need to be acquired to produce an image that has the same properties in terms of its reconstructed appearance as if all the data were acquired, a factor of two in time can be saved. On the other hand, if resolution rather than time is the issue, then the same number of data points as previously acquired can be used to create a data matrix twice as large as before, thereby increasing the resolution by a factor of two. In practice, however, the data acquired is never purely real due to artifacts and noise which are invariably present. As such, the images produced by imposing complex conjugate symmetry are only approximations of the images that would have been produced by collecting all of the data for the given data matrix.
In the case of magnetic resonance imaging, reconstruction of a given data matrix, or k-space as it is known in the field, from partially collected data has the potential of improving acquisition speed or resolution as described generally above. Prior art methods of data extrapolation, known as partial or fractional k-space coverage techniques, are well known in the field of magnetic resonance imaging and are set forth in detail in Liang et al., Constrained Reconstruction Methods in MR Imaging, Reviews of Magnetic Resonance in Medicine, Vol. 4, 67-185 (Pergamon Press Ltd. 1992). These methods have included one dimensional partial Fourier methods and constrained reconstruction techniques as described below. While the former can reduce imaging times by at most a factor of two, the savings in time from the latter methods is less predictable and is often less than a factor of two.
Initial attempts at using partial Fourier imaging utilized methods that simply acquired half of the k-space of a given data matrix and then used complex conjugation to recover the missing half. Typically, for a two-dimensioned image acquisition scheme, such methods involve acquiring data s(kx, ky) for the entire top half of k-space as shown in FIG. 1. In FIG. 1, the k-space shown is two-dimensional data where the two dimensions can represent the read and phase encoding directions or the phase and partition encoding directions. Assuming complex conjugate symmetry (i.e., s(xe2x88x92kx,xe2x88x92ky)=s*(kx, ky)), data for the lower half of k-space can then be approximated. This is performed using a one-dimensional approach by first transforming the acquired data along the horizontal direction to create a set of one-dimensional image/data set s(x, ky) for each x-position in the image matrix as shown in FIG. 2. With the data so transformed, the one-dimensional data in ky is then complex conjugated for each x-position to produce the missing half of the image matrix. However, as discussed above, such approaches are based on the assumption that the data are purely real and it is well known that complex conjugation cannot produce the missing half of k-space if the object is complex or when other sources of error are present in the data.
For the above reasons, a method was sought to remove any background variations in the raw data caused by the presence of imaginary information in the image itself. Such an attempt was first made by Margosian as disclosed in Faster MR Imagingxe2x80x94Imaging with half the Data, SMRM Conference Abstracts, Vol. 2, 1024-25 (1985). This method is well known in the art and is performed by extending the region of acquired data (usually by 8 points for spin echo data) as shown in FIG. 3 to obtain data for both positive and negative k-space points (i.e., before or after the origin in a given direction for an equal number of points). A low pass estimate of the phase, xcfx86(x, y), of the image is determined by using only the central portion of the acquired k-space data. Additionally, the acquired data is transformed as described in the above half k-space method after zero filling the remainder of the data matrix and applying an asymmetrical Hamming-like filter to the central portion of the data. This transformation produces a first estimate of the image that is complex, not real. An assumption is then made that the phase (xcfx86(x, y)) errors are small enough that the reconstructed image can be phase corrected by multiplying the first complex estimate of the image by exe2x88x92ixcfx86(x, y). The final image is presumed to be the real part of this second estimate. There are several disadvantages of using this approach. First, if the phase has any high spatial frequency components in it, the method will fail because too few points are used to estimate the phase. Increasing the number of points used would require collecting additional data, thereby reducing the time benefits of the method. Second, the filter used suppresses low spatial frequency information in the image itself, leading to an unnecessary loss in signal-to-noise over and above that inherent in the partial Fourier method itself. Finally, using solely the real part of the image matrix is not representative of the pristine image. Attempted methods of improving the phase estimate have not obviated these difficulties (SR McFall et al., Corrections of Spatially Dependant Phase Shifts for Partial Fourier Imaging, Magnetic Resonance Imaging, Vol. 6, 143-55 (1988)).
A similar approach using an iterative scheme which attempts to force the image to be real was developed by Cuppen et al. as disclosed in U.S. Pat. No. 4,853,635, the disclosure of which is incorporated herein by reference. Extensions to this approach have also been developed by Haacke et al. as disclosed in A Fast, Iterative, Partial-Fourier Technique Capable of Local Phase Recovery, Journal of Magnetic Resonance, Vol. 92, 126-45 (1991) which is incorporated herein by reference.
In general, the Cuppen approach has the advantage over the Margosian method in its ability to handle phase errors and to do so without an increased reduction of the signal-to-noise ratio above the sqrt(2) expected reduction. In Cuppen""s method, a similar one-dimensional approach is used to generate a complex image xcfx81(x). A new complex image, xcfx811(x), is then calculated as xcfx811(x)=xcfx81*(x)ei2xcfx86x. Estimated data for the k-space is then calculated by transforming xcfx811(x), thereby generating s1(kx). The estimated data, s1(kx), is then used in place of the uncollected data and is merged with the acquired data s(kx) to generate s1new(kx) which can then be transformed to generate a new complex image xcfx812(x), where xcfx812(x)=Fxe2x88x921(s1new(kx))*ei2xcfx86x (where F=1 represents the inverse Fourier transform operation). The above mentioned steps can then be iterated until a desired convergence has been obtained. This method, like other prior art methods, has several disadvantages. First, it has the potential to generate the correct or exact image, as opposed to an approximate image, only when the image is in fact real.
Second, as more points are added to improve the phase information, more than half of the k-space data must first be acquired, making the method not truly a half-Fourier method and reducing the expected savings in data acquisition time and effort.
Other methods also exist to try and extrapolate missing data using parametric estimation techniques as disclosed in Z. -P. Liang et al., Phase-constrained Data Extrapolation Method for Reduction of Truncation Artifacts, JMRI, Vol. 1, 721-24, (1991). Several attempts in this direction utilize models to change the basis functions of the data representation so that an entirely different reconstruction algorithm is used without involving Fourier transforms at all. These methods have many difficulties and are not in widespread use. One disadvantage is that the data reconstruction time for the methods are very long compared to the more common Fourier methods (often hundreds of times longer). Additionally, these methods are very sensitive to the shape of the structures being imaged and often produce very poor estimates of the image.
All Fourier methods of image reconstruction in the prior art, both single step methods (Margosian, McFall et al., etc.) and iterative methods (Cuppen, Haacke), are based on the above described one dimensional implementation and complex conjugation assumptions. As described above, to produce an acceptable image using these methods requires extending the acquired data near the central part of k-space to include at least a portion of symmetrically acquired data. This combined with the fact that complex conjugation only reflects data through the origin means that parts of k-space will have no estimates for their data points unless more than half the data points are originally acquired. Thus, these methods fail to reduce acquisition time by a factor of two as is desired. However, such methods have been utilized in the prior art for over 15 years and it has long been accepted that complex conjugation is required to solve the problem.
Contrary to the long felt belief that complex conjugation is required to reconstruct a partially acquired multidimensional data matrix, the method of this invention, although it can be used in conjunction with complex conjugation methods, utilizes an iterative multidimensional reconstruction technique that eliminates the need and disadvantages associated with using a complex conjugation method. Furthermore, the available data used to constrain the reconstruction increases as the number of dimensions increases while still maintaining the same time savings of a factor of two. This provides a significant improvement in image quality over prior art methods.
Generally, the method of this invention comprises calculating a magnitude component (xcfx81m1(x, y)) of the image matrix by transforming the data matrix s(kx,ky) using a multidimensional reconstruction technique such as a multidimensional inverse Fourier transform. This magnitude component can then be merged with a phase component (xcfx86e(x, y)) of the image matrix, which like the magnitude component is calculated by transforming the data matrix using a multidimensional reconstruction technique, but with the data matrix filled with less than all the collected data or with image data information from a second sampling. By merging the magnitude and phase components from two separate transformations, an image matrix xcfx811(x, y)=xcfx81m1(x, y)eixcfx86e(x, y) is created which can then generate estimated values for the data matrix s1(kx,ky) by transforming the image matrix back into the data matrix domain using a multidimensional reconstruction technique such as a multidimensional forward Fourier transform. With the data matrix filled with the collected data in its original form and with estimated data values filling the uncollected portion of the data matrix, a new magnitude component xcfx81m2(x, y) can be calculated by again transforming the data matrix s1(kx,ky) using a multidimensional reconstruction technique. Thus, an iterative process results using the phase component as a constraint whereby the new magnitude component is combined with the original phase component to form a new image matrix xcfx812(x, y)=xcfx81m2(x, y)eixcfx86e(x, y) which, if necessary, can be used to generate new estimated data values yet again, and so forth until convergence is reached.
In principle, this method is complementary to the usual partial Fourier imaging and, in fact, can be used in place of it to generate unacquired data and save a factor of two or more in acquisition time. As in parametric methods, the invention need only acquire as many data points as there are unknowns. Using the method of this invention, no complex conjugation is needed and the data collected can be designed to save exactly a factor of two in time, while still obtaining a much larger coverage of the central part of k-space than previous methods to ensure that the phase data is very accurately determined. Furthermore, the method does not require many of the assumptions necessary with previous one-dimensional methods and, as such, it is a more robust method than those previously used.